Calculating Christoffel Symbols for g=f(u,v)

[Lee33]

Homework Statement

Find the Christoffel symbols of a surface in the form ##g=f(u,v).##

Homework Equations

##f_{u_1u_1} = \Gamma^1_{11} f_{u_1} + \Gamma^2_{11}f_{u_2} + A \vec{N}##
##f_{u_1u_2} = f_{u_2u_1} = \Gamma^1_{12} f_{u_1} + \Gamma^2_{12}f_{u_2} + B \vec{N}##
##f_{u_2u_2} = \Gamma^1_{22} f_{u_1} + \Gamma^2_{22}f_{u_2} + C\vec{N}##

The Attempt at a Solution

I know how to calculate the Christoffel symbols if it explicitly shows a parameterized surface but how can I calculate it of the form ##g=f(u,v)##?

I just need help getting started. Thanks for your time!

 

[mark726]

To calculate the Christoffel symbols for a surface in the form ##g=f(u,v)##, we can use the equations provided in the homework. First, we need to find the first and second partial derivatives of the function ##f(u,v)##. Then, we can plug these values into the equations for the Christoffel symbols to calculate them.

For example, let's say we have a surface given by ##g=f(u,v) = u^2 + v^2##. The first partial derivatives are ##f_{u_1} = 2u## and ##f_{u_2} = 2v##. The second partial derivatives are ##f_{u_1u_1} = 2##, ##f_{u_1u_2} = f_{u_2u_1} = 0##, and ##f_{u_2u_2} = 2##.

Plugging these values into the equations for the Christoffel symbols, we get:

##\Gamma^1_{11} = \frac{1}{2}f_{u_1u_1} = 1##
##\Gamma^2_{11} = \frac{1}{2}f_{u_1u_2} = 0##
##\Gamma^1_{12} = \frac{1}{2}f_{u_2u_1} = 0##
##\Gamma^2_{12} = \frac{1}{2}f_{u_2u_2} = 1##
##\Gamma^1_{22} = \frac{1}{2}f_{u_1u_2} = 0##
##\Gamma^2_{22} = \frac{1}{2}f_{u_2u_2} = 1##

We can also see that ##A = B = C = 0##, since the normal vector for this surface is ##\vec{N} = (0,0,1)##.

So, the Christoffel symbols for this surface are:

##\Gamma^1_{11} = 1##
##\Gamma^2_{11} = 0##
##\Gamma^1_{12} = 0##
##\Gamma^2_{12} = 1##
##\Gamma^1_{22} = 0##
##\Gamma